Despite already having a few years of professional experience dealing with Linux servers, I still, to this day, confuse, which environment file gets sourced and when...

There's /etc/profile, /etc/bashrc, ~/.bash_profile, ~/.profile, ~/.bashrc

I think it's... Bashrc for interactive shells, profile for login shells.

But then I have examples like "ssh user@server 'echo $var'" that... Don't source any of the files!

You can enable user environment files for SSH that get sourced whenever a user logs on through SSH (~/.ssh/environment / environment specified for a key in ~/.ssh/authorized_keys)

Is there some sort of master environment file that gets sourced *every* time, no matter what kind of shell starts?